The class of processes that we are studying in this paper were introduced and studied in [1]}. As we mentioned before, each of the \((S^{{{\scriptscriptstyle {({k}})}}}_n)_n\)  with \(k \ge 1\) , is an example of process with long memory. The process \(S_n^{{{\scriptscriptstyle {({1}})}}}\)  has many similarities, in terms of construction, with the Elephant Walk, which can be described as follows. Elephant random walks captured a lot of attention in the literature in recent years, as they are long-memory processes connected with random recursive trees, and exhibit anomalous diffusion properties. This family of processes have a parameter \(p\)  and are built as follows. The first row of the array is built in the same way the BRW is assembled, i.e. \(\mathcal {E}_{1, i} = \xi _i\)  where the \(\xi \)  are i.i.d. in \(\lbrace -1, 1\rbrace \)  with mean \(2p -1\) .
Let \((U_k)_k\)  be a sequence of independent random variables, where \(U_k\)  is discrete uniform over the set \([\![1,k]\!]\) . Fix \(\mathcal {E}_{2, 1} = \mathcal {E}_{1, 1}\)  and suppose defined \(\mathcal {E}_{2, n}\)  with \(n \in \mathbb {N}\) . We define \(\mathcal {E}_{2, n+1}\)  as \(\xi _{n+1}\mathcal {E}_{2, U_n}\) . Define \(E_n = \sum _{j=1}^n \mathcal {E}_{2, j}\) . The process \((E_n)_n\)  is known in the literature as an Elephant Random Walk and was introduced by Schütz and Trimper [2]}. It was studied, among others, by Baur and Bertoin [3]} where central limit theorems are proved for a certain range of values of \(p\) , and a convergence to non-Gaussian limits for other values. See also Gonzales, González-Navarrete and Lambert [4]}.
